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Wall roughness induces form-induced (or dispersive) velocity and pressure perturbations inside the roughness sublayer of a wall-bounded turbulent flow. This work discusses the role played by the form-induced velocity in influencing turbulence statistics and structure , using existing direct numerical simulation data of transient half channels in response to an impulse acceleration (Mangavelli et al., J. Turbul., 22:434-460, 2021). Focuses are given to (i) reshaping of turbulent coherent motions by the rate-of-strain of the mean velocity, and (ii) contributions of different velocity sources to turbulent pressure fluctuations. Half-channel flows in both fully-developed and non-equilibrium, transient states are discussed. Results show that form-induced velocity gradients not only form an important source of turbulent pressure in an equilibrium flow, but also lead to turbulence production and potentially direct structural change of turbulent eddies in a non-equilibrium flow under acceleration.
Accepted for publication in Journal of Turbulence 1
Eects of form-induced velocity in rough-wall
turbulent channel ows
By S. C. Mangavelli and J. Yuan
Wall roughness induces form-induced (or dispersive) velocity and pressure perturba-
tions inside the roughness sublayer of a wall-bounded turbulent ow. This work discusses
the role played by the form-induced velocity in inuencing turbulence statistics and struc-
ture, using existing direct numerical simulation data of transient half channels in response
to an impulse acceleration (Mangavelli et al.,J. Turbul.,22:434460, 2021). Focuses are
given to (i) reshaping of turbulent coherent motions by the rate-of-strain of the mean
velocity, and (ii) contributions of dierent velocity sources to turbulent pressure uctua-
tions. Half-channel ows in both fully-developed and non-equilibrium, transient states are
discussed. Results show that form-induced velocity gradients not only form an important
source of turbulent pressure in an equilibrium ow, but also lead to turbulence produc-
tion and potentially direct structural change of turbulent eddies in a non-equilibrium
ow under acceleration.
1. Introduction
Wall roughness plays an important role in many elds of study. Much has been done
to identify the eects of roughness on wall-bounded turbulence, equilibrium or non-
equilibrium, with focuses on wall friction and statistics of mean ow and turbulence. See
reviews provided by Raupach et al. (1991), Jim´enez (2004), Chung et al. (2021), as well
as Finnigan (2000) in atmospheric applications. In addition, recent reviews on current
knowledge regarding rough-wall non-equilibrium turbulence were provided by Devenport
& Lowe (2022) and Volino et al. (2022).
One approach to describe and quantify the dynamical eects of roughness on turbulence
is to analyze changes in the governing equations due to the presence of roughness. As
shown by Raupach & Shaw (1982), Mignot et al. (2009) and Yuan & Piomelli (2015),
additional terms in the transport equations of linear momentum and Reynolds stresses
appear with the presence of roughness when a double-averaging process is applied to the
equations. This averaging process performs a triple decomposition of an instantaneous
uid variable θ(x, y, z, t), such that θ=θ+!
θ+θ, where () is time averaging for a
steady problem (or ensemble averaging for an unsteady one), ()= () () is turbulent
uctuation, and is intrinsic spatial averaging (i.e. value per unit uid area) along
homogeneous directions. !
θ=θ θis the spatial heterogeneity of the local averaged θ,
called the dispersive or form-induced uctuations. Supercial spatial averaging ·s(i.e.
uid variable per unit total area) is also used. θis the double-averaged (DA) value.
Department of Mechanical Engineering, Michigan State University
Corresponding author. Email:
2Mangavelli & Yuan
The DA linear momentum equation is
Roughness gives rise to two additional terms in Equation (1.1): the fourth term on
the right-hand-side representing the momentum transfer due to form-induced stresses,
!ui!ujs(Nikora et al. (2001), also termed dispersive stresses by Wilson & Shaw (1977)),
and a body force (fi) representing the sum of local pressure drag and viscous stress
imposed on the surface of roughness. These two terms are signicant near the wall.
The near-wall layer in which the form-induced stress is signicant compared to the DA
velocity is called the roughness sublayer (RSL) (Pokrajac et al. 2007). Jelly & Busse
(2019) showed that both !u!vand uv, in opposite signs, contribute signicantly to the
mean momentum balance near the roughness crest for a Gaussian roughness. Mangavelli
et al. (2021) showed that, although in a fully developed channel ow !u!vis much weaker
than uv, in a transient channel ow the !u!vmagnitude may exceed temporarily that
of uvbelow the roughness crest.
The DA equations of normal Reynolds stresses (considering a channel ow) can be
written as (no summation over Greek index),
% &' (
% &' (
% &' (
% &' (
% &' (
% &' (
% &' (
% &' (
where the terms on the right-hand-side are, respectively, shear production due to double-
averaged strain rates (Ps), additional shear production due to form-induced strain rates
s), additional transport due to form-induced velocities (Tw), turbulent transport (Tt),
viscous transport (Tν), pressure-strain-rate term (Π), pressure transport (Tp), and viscous
dissipation (%). Discussions on how the form-induced velocity modies Reynolds-stress
balance focused mainly on the Pf
sterm, which represents conversion from the kinetic
energy of the form-induced uctuations at the roughness length scale (or wake kinetic
energy, WKE (Raupach & Shaw 1982)) to the turbulent kinetic energy (TKE) at smaller
scales. The form-induced production was found to depend on the roughness geometry
(Yuan & Aghaei-Jouybari 2018) and to play a more important role in Reynolds stress
balance in non-equilibrium ows, such as in spatially accelerating boundary layers (Yuan
& Piomelli 2015), oscillatory channel (Ghodke & Apte 2016) and transient accelerating
channels (Mangavelli et al. 2021), than in canonical ows (Mignot et al. 2009; Yuan &
Piomelli 2014c). In these studies, Twwas found to be small in comparison to other terms.
Besides TKE production, another key process that controls the development of tur-
bulence is the distribution of TKE among velocity uctuations in dierent directions,
represented by the pressure strain term in Equation (1.2) and partially controlled by
local pintensity. It is not yet clear whether and how form-induced velocity modies
turbulent pressure and TKE redistribution.
Rough-wall turbulent ows in engineering applications such as those over naval vessels
Eects of form-induced velocity in turbulent channel ows 3
or airfoils/hydrofoils usually display temporal and/or spatial variations due to external
factors including freestream pressure gradient, surface curvature, and ow unsteadiness.
These ows are termed non-equilibrium if a similarity solution of velocity cannot be
found. One type of non-equilibrium turbulence widely studied is the one under accelera-
tion (or favorable longitudinal pressure gradient, FPG). Studies on smooth walls showed
that a strong freestream acceleration causes the boundary layer to undergo reverse tran-
sition from a fully turbulent state to a quasi-laminar state (Narasimha & Sreenivasan
1973). This was attributed to the stretching of near-wall turbulent eddies, leading to elon-
gated velocity streaks, quasi-one-dimensional turbulence, reduced TKE redistribution,
and consequently stabilized turbulence, whose intensity decouples from the accelerated
mean ow (Bourassa & Thomas 2009; Piomelli & Yuan 2013; Volino 2020). Roughness
acts to counter the stabilizing eects of FPG (Cal et al. 2009; Yuan & Piomelli 2015), as
it augments near-wall TKE and leads to a more isotropic Reynolds stress tensor.
Transient accelerated ows were shown to be fundamentally similar to those that un-
dergo spatial acceleration. Based on DNS, He & Seddighi (2013) and Seddighi et al.
(2015) characterized the response of turbulent channel ows to an impulse-like increase
in ow rate, over smooth and rough walls. On the smooth wall, they observed similar ow
response as that in a FPG boundary layer undergoing reverse transition: elongated low-
speed streaks, laminar-like mean velocity proles, higher Reynolds stress anisotropy, and
a frozen pressure strain term. In the presence of pyramid roughness, such reverse transi-
tion was prevented. Based on DNS of transient half-channel ows with a similar congu-
ration, Mangavelli et al. (2021) characterized how two dierent roughnesses with similar
average and root-mean-square heights but dierent geometries aect the evolutions of
mean and turbulent statistics. !uiin the RSL was observed to stay quasi-equilibrium
(scaling with the velocity at the edge of the sublayer) throughout the transient process,
in stark contrast to the non-equilibrium variation of u
i. Strong form-induced production
of TKE in all u
icomponents contributes to a much faster recovery of the steady state
on rough walls than on the smooth wall. The roughness geometry determines how fast
turbulence recovers to the steady state.
The description above summarizes the current knowledge on how roughness, as well
as the !uields it induces, dynamically aects equilibrium and non-equilibrium wall-
bounded turbulent ows. However, a few questions are still unanswered. One is whether
form-induced velocity meaningfully aects Reynolds stress balance in a way other than
additional TKE production, for example in modifying turbulent pressure and TKE redis-
tribution. In addition, most previous attention were given to eects on turbulent statis-
tics. Does !uialso contribute to dierent characteristics of turbulent structure observed
on dierent rough walls?
2. Objectives
This work investigates the above questions in equilibrium and non-equilibrium channel
ows, based on existing DNS data (Mangavelli et al. 2021) of transient half-channel ows
subject to impulse accelerations on a smooth wall and two rough walls: a homogeneous,
densely distributed sandgrain roughness and an inhomogeneous, multiscale roughness
obtained from a hydraulic turbine scan. Mangavelli et al. (2021) focused on the devel-
opment of turbulent statistics, while the present work extends the analyses to turbulent
structure and various pathways through which !uiaects turbulence.
The organization of the rest of the manuscript is as follows. In Sec. 3, simulation
4Mangavelli & Yuan
Case Reb0Reτ0k+
SM 4,000 244 0 (4.0, 2.0) 12,000 626 0 (9.8, 4.8)
S 4,000 320 21 (5.0, 2.5) 12,000 1023 76 (15.3, 7.6)
T 4,000 294 10 (3.4, 3.4) 12,000 858 23 (10.1, 10.1)
Table 1: Simulation parameters at initial (0) and new (1) steady states. + indicates
normalization in wall units (i.e. uτand viscous length scale δν=ν/uτ). xiis the cell
size in xidirection. Reb=ubδ/νand Reτ=uτδ/ν.δis channel half height.
methodologies and parameters are described; statistical comparisons of the ow are also
summarized. Then, two-point velocity correlations and spectra are compared in both
steady and transient states in Sec. 5.1. To provide an explanation to the structural
dierence, the rate-of-strain tensor of !uiis characterized in Sec. 5.2. The velocity sources
of the turbulent pressure are compared in Sec. 5.3 to evaluate the eects of ui,!uiand
ion pressure uctuations. Conclusions are provided in Sec. 6.
3. Problem fomulation
A summary of the methodologies and simulation setups as reported in Mangavelli et al.
(2021) is provided here. The readers are referred to that publication for further details.
The incompressible ow of a Newtonian uid was simulated by solving the equations
of conservation of mass and momentum. x,yand zare, respectively, the streamwise,
wall-normal and spanwise directions of the half-channel ow, and u,vand ware the
velocity components in those directions. pis the pressure, ρis the density and νis
the kinematic viscosity. Periodic boundary conditions are applied in xand zdomain
boundaries, and symmetric and no-slip conditions are applied to the top and bottom
boundaries, respectively. In the rough cases, an immersed boundary method was used
to impose the no-slip and no-penetration boundary condition on the rough walls. It is
based on the volume-of-uid approach; its detailed implementation in the in-house uid
solver was described by Yuan & Piomelli (2014c) and Yuan & Piomelli (2014b). The
governing equations were solved on a staggered grid using second-order central dierences
for all terms, second-order accurate Adams-Bashforth semi-implicit time advancement,
and Message-Passing Interface parallelization.
To conduct the present transient channel simulations, rst a separate simulation was
carried out for each case at the initial steady state with a bulk Reynolds number Reb0,
to generate statistics at this state and be used as initial conditions for the transient
simulations. The transient simulations were then carried out, based on independent initial
conditions, for around 20 times for each case to collect data for ensemble averaging at each
time t. During a transient simulation, the variation of Reb(t) was imposed, equivalent
to a rapid three-fold linear increase of the bulk velocity (ub) that started at time t= 0
and lasted for a duration of t=tub0/δ= 0.08 (see Figure 2a). ubremained constant
thereafter. This simulation setup is similar to that of He & Seddighi (2013), though with
quantitative dierences in the Reynolds number and the amount and rate of the velocity
Simulation parameters for all cases are listed in Table 1. The initial and nal steady
states are denoted using subscripts 0 and 1, respectively. SM, S, and T repre-
sent the smooth-wall, sandgrain, and a multiscale turbine-blade-roughness cases, respec-
Eects of form-induced velocity in turbulent channel ows 5
Figure 1: (a) Geometries of sandgrain (case S) and turbine-blade-roughness (case T)
surfaces colored by height, with zoomed-in view. (b) Power spectral density of height
uctuations, with wavenumber κin x(black) and z(gray). (c) Sketch of a rough surface
showing various length denitions.
tively. The same time-variation of Rebwas imposed in all cases. Both rough cases started
in transitionally rough regime at the initial steady state and became fully rough at the
new steady state, as shown by the k+
svalues. Here, ksis the equivalent Nikuradse
sandgrain height in the fully rough regime of a rough surface; the critical values of k+
for the present roughnesses corresponding to the lower limit of the fully rough regime
were reported by Yuan & Piomelli (2014a).
On a rough wall, the wall shear stress τwresults from both the viscous and pressure
drag due to roughness. It was calculated as
τw(t) = ρ
f1(x, y, z, t) dxdydz+µus
6Mangavelli & Yuan
Surface Ra/δkc/δkrms/RaskkuESxESzyR/kc
S 0.014 0.09 1.05 0.48 2.97 0.43 0.44 1.10 ±0.13
T 0.014 0.13 1.17 0.20 3.49 0.10 0.08 1.05 ±0.10
Table 2: Characteristics of the rough surfaces.
where f1(x, y, z, t) is the streamwise component of the immersed boundary method body
force, Vis the volume of the simulation domain, and Lxiis the domain length in xidirec-
tion. For the two rough surfaces considered herein, the second term on the right-hand-side
of Equation (3.1) is negligible. The friction velocity was then obtained as uτ(t) = 2τw/ρ.
The corresponding friction Reynolds numbers are tabulated in Table 1; the sandgrain
roughness (case S) produced higher wall friction than the multiscale roughness (case T).
Grid sizes of the uniform mesh in xand zare shown in wall units in Table 1. In y, the
grid is rened near the wall. At the nal steady state (i.e. the more critical one for spatial
resolution between the two steady states), y+
min <0.3 for SM and between 0.6 and 1.7
in the layer below roughness crest for the rough cases.
Two dierent surfaces roughness, one synthetic sandgrain roughness S and one repli-
cated from a surface scan on a hydraulic turbine blade T, are used in the simulations.
Figures 1(a) and (b) compare the roughness geometries and the power spectral density
of their height uctuations. Case S shows a prominent spectral peak at the separation
length between neighboring grains in xand z, denoted by λs, while the T roughness re-
sembles a fractal roughness with a spectral power decay of 2, without a dominant peak
at a particular length scale. For further discussion regarding the height power spectral
densities, see Mangavelli et al. (2021).
Parameters of the two rough surfaces are compared in Table 2. Both surfaces share the
same rst-order (Ra) and similar second-order moments (krms) of height statistics, but
quite dierent values of maximum peak-to-trough height (kc, also called crest height),
skewness (sk) and eective slope (ES) in xor z. Both surfaces are positively skewed (i.e.
peaky), with kurtosis values of around 3. Compared to S, the T roughness displays sparser
distribution of peaks, leading to a lower skewness value and milder eective slopes. The
thickness of the RSL (i.e. yR, dened as the layer in which !u21/2(y)/u(y)>0.1) is
around 1.1 times of kcin both cases. yRvaries mildly (within around 10%) throughout the
transient process. The domain lengths in xand zare 12δand 6δ, respectively, for cases
SM and S, and both 12δin case T to accommodate its large in-plane roughness length
scales in both xand z. For case S, the spatial resolution of the grain geometry is 8 and
16 points per grain in xand z, respectively. For case T, there is no dominant roughness
length scale. The Taylor micro-scale (λT) (Yuan & Piomelli 2014a; Aghaei Jouybari
et al. 2021) is thus used to quantify the length of an equivalent roughness element. λTis
resolved by around 10 points in both xand z.
4. Summary of turbulence statistics
Some observations on ow statistics are summarized here to provide a context for the
structural and other analyses reported in Section 5. For comprehensive discussions of the
ows, see Mangavelli et al. (2021).
Figure 2(a) shows the acceleration of ubthat was imposed in all cases. The friction co-
Eects of form-induced velocity in turbulent channel ows 7
Figure 2: Flow statistics of the transient channels: (a) bulk velocity, (b) friction coecient,
and wall-normal maximum values of streamwise and vertical normal Reynolds stresses
(c,d) and form-induced stresses (e,f). (b-f) are reproduced from Mangavelli et al. (2021)
with permission.
ecient Cf= 2τw(t)/(ρu2
b0) is compared in Figure 2(b). Cfundergoes a sudden increase
for all cases immediately following the ubramp-up, and decreases rapidly as the ow
starts to recover. Figures 2(c-f) highlight the dierences in Reynolds and form-induced
stress developments in the three cases.
On the smooth wall, turbulence undergoes reverse transition toward a quasi-laminar
state, characterized by a dip of Cfbelow its long-time value and high Reynolds stress
anisotropy (with TKE mostly resting in u), caused by a delayed response of the TKE
redistribution to the acceleration. On the rough walls, however, a clear dip of Cfis absent,
8Mangavelli & Yuan
Figure 3: Contours of R11 at levels 0.3(0.2)0.9 in Cases SM (a), S (b) and T(c), at four
time instances: t= 0.0, 0.5, 5, and 22. Lines tted based on outermost points (an
example is shown with in (a)) at contour levels 0.5(0.1)0.9, to show inclination angles
(marked in red). For clarity, a shift of 4.5 and 1.5 units are used for smooth and rough
cases, respectively.
and Reynolds stress anisotropy stays almost unchanged. As opposed to the monotonic
increase of u
iuctuations to a higher-Reynolds-number state, the !uiuctuations display
rapid augmentation following the ubramp-up and then decrease toward the long-time
values. This variation of !uiintensity was shown to be a result of the variation of uat
the edge of the RSL; the ratio between the root-mean-square (RMS) of !uiand u(yR)
was shown to be almost constant for each rough case.
5. Results
5.1. Structural characteristics of turbulent velocity
First, the two-point auto-correlation of the turbulent velocity u
icentered at an elevation
yref is calculated as
R11(rx, ry, t) = u(x, yref , z, t)u(x+rx, yref +ry, z, t)
u2(yref , t),(5.1)
where rxand ryare separations in streamwise and wall-normal directions, respectively.
The intrinsic averaging operator in the nominator on the right-hand-side of Equation (5.1)
indicates that only contributions from products with both points (*xand *x+*r) in the
uid domain are accounted for in calculating the correlation. Contours of R11 in the
(x, y) plane are shown in Figures 3 for all cases at t= 0.0, 0.5, 5, and 22. In the smooth
case, the correlation is centered at y+
ref,0=uτ0yref/ν= 10, which corresponds to the
peak elevation of the TKE production at t= 0, while in the rough cases yref/kc= 1,
which is near the peak elevations of normal Reynolds stresses (Mangavelli et al. 2021).
Following Volino et al. (2007), the inclination angle θis calculated using the best tted
line by the linear least square method, based on the points farthest away from the self-
correlation point at (rx, ry) = (0,0), both upstream and downstream on the contour lines
Eects of form-induced velocity in turbulent channel ows 9
Figure 4: Temporal variation of streamwise integral length (a,c) and inclination angle
(b,d) based on R11 in SM () , S (), T ( ) cases. The correlations are centered at
ref,0= 10 for SM and yref/kc= 1 for S and T. (a,c) show linear scalings of tand (c,d)
show logarithmic scalings.
with levels from 0.5 to 0.9. These outermost points used for line tting are shown in the
R11 contours at t= 0.0 in Figure 3(a), as an example. The values of θare listed in the
gure. To compare quantitatively the size and inclin